What if Charlemagne does not divide his empire after death? I am not that experienced in frankish history and culture to assess, if thats possible.
Charlemagne didn't divided his empire after his death, he tried to do so in 806, but his son was the sole inheritor.
Louis tried to not divide the empire, at least make the others sons submitting to an imperial suzerainty of Lothar, but it failed with much civil war.
That's only a part of the problem, but contradictions between frankish customs (equal division of royalty) and tentatives of actual imperial sucessions were too importants and benefitted too much to the carolingian elites to be that sucessful.
Let's admit, for the sake of the discussion, that the empire remains united (only one surviving heir by exemple). Carolingia is still pretty much challenged.
This picture resumes quite well some problems it faced.
Others factors clearly made the crisis happen and can't be limited to Islamic overspending : by exemple, Vikings raids due to a possible demographical crisis (Marc Bloch speculative thesis), no more possibility of fructuous raids redistributing wealth among frankish nobility (forced to improve their production, and a more localist stance), collapse of Frisian takeover of North Sea trade replaced by Norses, etc.
Admitting Carolingia survives as a whole, their possibilities of expansion are pretty limited : Spain is, while divided, pretty much under Cordoban dominance (while itself on crisis) and crossing (with which fleet?) a North Sea or a Channel virtually under Viking dominance seems unlikely.
I would think Carolingians would have enough trouble on their hands, trying to stay in power to begin with (something they didn't managed to do OTL), stand united and eventually strong enough to launch such invasions.
It's not impossible to do, but would require much change.